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<span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c4">G</font> st <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">G9</font> = <font color="Maroon" title="c4">G</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">h</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> holds  <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">g</font></span>)</span></span><br/><a NAME="E1:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E2:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font> as   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a>, <a class="ref" href="group_6.html#T38" title="GROUP_6:th.38">GROUP_6:38</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E4:45_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E5:45_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:45_1_1"/>
<font color="Maroon" title="c7">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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